The question concerns a special case of "Hindman's theorem":
Suppose that the natural numbers are colored with $r$ different colors.
Then there exists a color $c$ and an infinite set $D$ of natural numbers, so that all elements of $D$ are colored with $c$ and so that every finite sum of elements of $D$ also has color $c$.
For solving the puzzle, you just apply Hindman's theorem with $r=3$ colors (red, yellow, green). The theorem gives you an infinite mono-chromatic set $D$, from which you may pick three arbitrary elements $a,b,c$; these three elements have all desired properties.
- A wikipedia article on so-called IP sets and Hindman's theorem
- The article "A Simple Proof and Some Difficult Examples for Hindman's Theorem" by Henry Towsner on the arXive
- Example 3 in the Tricki article "How to use ultrafilters" contains another proof of Hindman's theorem